自由阿尔贝群中的球面几何
The geometry of spheres in free abelian groups
摘要：
我们通过开发足够精细的工具来测量对生成集的依赖性来研究Z^d上的词度量。我们得到了长度为n的字的计数和分布结果。由此，我们证明了球面上的计数测度总是收敛于极限形状上的极限测度（在适当的意义上是强收敛的）。极限测度的存在性是相当强的--甚至实际上阿贝尔群也不需要满足这类渐近公式。利用极限测度，我们可以把关于词度量的概率问题归结为欧几里得空间的凸几何问题。作为一个应用，我们给出了关于任何生成集的球形增长函数的渐近性，以及其他“大小”函数的统计。
英文摘要：
We study word metrics on Z^d by developing tools that are fine enough to measure dependence on the generating set. We obtain counting and distribution results for the words of length n. With this, we show that counting measure on spheres always converges to a limit measure on a limit shape (strongly, in an appropriate sense). The existence of a limit measure is quite strong-even virtually abelian groups need not satisfy these kinds of asymptotic formulas. Using the limit measure, we can reduce probabilistic questions about word metrics to problems in convex geometry of Euclidean space. As an application, we give asymptotics for the spherical growth function with respect to any generating set, as well as statistics for other "size-like" functions.
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